vorschieben
to push forward
To move something forward by pushing it.
Er schob den schweren Riegel vor die Tür.
He pushed the heavy bolt in front of the door.

to push forward / use as a pretext
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to push forward
To move something forward by pushing it.
Er schob den schweren Riegel vor die Tür.
He pushed the heavy bolt in front of the door.

'Vorschieben' (to push forward / use as a pretext) is a separable verb with the prefix 'vor-'. In a main clause, the prefix goes at the end; in a subordinate clause, the verb stays together. In the infinitive with 'zu', 'zu' goes between 'vor' and 'schieben'.
Er schiebt den Tisch einen Meter vor.
He pushes the table one meter forward.
The verb consists of 'vor' (forward) and 'schieben' (to push). The literal meaning is 'to push forward.' The figurative meaning (to use an excuse) is derived from this: you symbolically push a reason in front of yourself in order to hide behind it.
Um nicht aufräumen zu müssen, schob er seine Kopfschmerzen vor.
To avoid having to tidy up, he used his headache as a pretext.
In the figurative meaning, 'vorschieben' is often used with abstract nouns. Typical combinations are 'eine Ausrede vorschieben' (to use an excuse as a pretext), 'einen Grund vorschieben' (to put forward a reason as a pretext), or 'einen Vorwand vorschieben' (to use a pretext).
Er hat einen dringenden Anruf vorgeschoben, um das Meeting früher zu verlassen.
He used an urgent call as a pretext to leave the meeting early.
In the meaning 'sich vordrängeln' (to push ahead / cut in line), the verb is always used reflexively with 'sich'. The reflexive pronoun must be adapted to the person (ich schiebe mich vor, du schiebst dich vor, etc.).
An der Kasse hat sich einfach jemand vor mich vorgeschoben.
At the checkout, someone simply pushed in ahead of me.
to imagine / introduce
'vorstellen' and 'vorschieben' can both mean that something is moved forward. The difference lies in the type of movement. You use 'vorstellen' when you move something forward or bring it into a more forward position. 'vorschieben' is more precise when something is really moved forward by pushing, for example a chair or a table. Learners may confuse the words because both begin with 'vor-' and express movement forward. As a practical rule: if the pushing is important, 'vorschieben' is more precise.

Im Restaurant schiebt Tom den Tisch etwas vor, damit alle besser sitzen können.
In the restaurant, Tom pushes the table forward a little so that everyone can sit more comfortably.
Kannst du den Stuhl bitte ein Stück vorstellen?
Could you move the chair forward a bit, please?